Representations are useful in all areas of mathematics because they help us develop, share, and preserve our mathematical thoughts. "[They] help to portray, clarify, or extend a mathematical idea by focusing on its essential features" (NCTM, 2000, p. 206).
"Mathematical representation" refers to the wide variety of ways to capture an abstract mathematical concept or relationship. A mathematical representation may be visible, such as a number sentence, a display of manipulative materials, or a graph, but it may also be an internal way of seeing and thinking about a mathematical idea. Regardless of their form, representations can enhance students' communication, reasoning, and problem-solving abilities; help them make connections among ideas; and aid them in learning new concepts and procedures.
"Good representations fulfill a dual role: they are tools for thinking and instruments for communicating . . . Learning to record or represent thinking in an organized way, both in solving a problem and in sharing a solution, is an acquired skill for many students. Teachers can and should emphasize the importance of representing mathematical ideas in a variety of ways."
(NCTM, 2000, p. 206, 208)
Defining Representation "The ways in which mathematical ideas are represented is fundamental to how people can understand and use those ideas . . . When students gain access to mathematical representations and the ideas they represent, they have a set of tools that significantly expand their capacity to think mathematically."
(NCTM, 2000, p. 67)
Representations include symbols, equations, words, pictures, tables, graphs, manipulative objects, and actions as well as mental, internal ways of thinking about a mathematical idea. Representations are powerful thinking tools. However, for many students, this power is not accessible unless they receive purposeful guidance in expanding their repertoire.
The act of representing a concept or relationship may result in the use of manipulative materials, the construction of graphs or diagrams, the writing of number sentences, or the presentation of a written or oral explanation. When using representations to solve a problem or make sense of a new concept, students are likely to go back and forth, using the representation to help clarify the problem and using the problem to extend their understanding of the representation.
"Representations do not 'show' the mathematics to the students. Rather, the students need to work with each representation extensively in many contexts as well as move between representations in order to understand how they can use a representation to model mathematical ideas and relationships." (NCTM, 2000, p. 208)
Using Variety of Representations
Much of students' mathematical learning involves expanding understanding of a mathematical idea or relationship by shifting from one type of representation to a different representation of the same relationship. This is one of the reasons that it is important for students to use a variety of manipulative materials, which are then carefully related to paper-and-pencil methods of solving problems. Through this work, they move from informal representations to the more formal, and abstract representations that more advanced work will require.
A third-grade teacher introduced a unit of study on two-digit multiplication with an open-ended assignment for student pairs. The students were asked to think of a story problem where someone would want to know the product of 15 x 12 and to then show a method for finding that product:
Make up a story problem that would go with this number sentence: 15 x 12 = ?
Write your story problem.
Show a method for finding the product of 15 x 12.
Discuss your method with another group. What does your group's representation show differently? What is most clear?
Students suggested several different contexts and manipulative materials to be used. For example, one suggestion was 15 plastic bags with 12 crayons each (represented by 15 rectangles with "12 crayons" written on each rectangle). Another suggestion was a dot array of 15 teams lined up on the playground, with 12 players on each team. Another group of students worked first on their solution method and struggled to match 15 x 12 to a story problem. They used a method that had been used to introduce multiplication in the prior grade: making 15 towers of linking cubes with 12 cubes each and then finding an efficient method for counting them all:
15 Linking Cube Towers
The methods used for finding the product were somewhat dependent on the representation that was used. The group with the crayons added 12 plus 12 plus 12, etc. -- first mentally, then on paper -- to find the total; they had some difficulty keeping track of the number of times that 12 was added:
Student Writing
The two groups with the array of team members and with the towers of linking cubes both experimented with a variety of ways of finding the product before partitioning their rows of dots or towers of cubes into two parts, 10 and 2. They worked first with the rows or towers of 10, finding that partial product easily, and then found the partial product of 15 times 2.
Partitioned Student Dots
Partitioned Linking Cube Towers
As students invent their own way to show a relationship, such as 15 x 12, with materials, pictures, or diagrams, they engage in thought that helps strengthen their understanding of the operation of multiplication. When a number of different representations for a given problem are shared and discussed, similarities in the mathematical structure of each representation can be highlighted. For example, a student with 15 towers with 12 linking cubes in each tower can point out the similarity to a representation that uses 15 plastic bags with 12 crayons in each. Similarly, an area diagram can be related to an array that is made of 12 rows with 15 objects per row:
Generic Area Model
Notice that while an array representation may initially encourage students to use simple counting to solve the problem, an area model clearly shows the advantage of making smaller, easier to calculate groups or areas. This also very clearly corresponds to the standard algorithm for multiplication, as shown above.
During the following week, the teacher extended the example of the lines of team players and connected it to using base-ten blocks to represent 1, 10, or 100 players instead of drawing individual dots. Over the course of the next several weeks, through class discussion and guidance from the teacher, the class developed connections between this manipulative model, arrays drawn on grid paper, and symbolic methods for finding the product of two two-digit numbers. They also practiced mental-math methods for finding products by breaking a problem into two parts, such as (15 x 10) + (15 x 2).
"[D]ifferent representations support different ways of thinking about and manipulating mathematical objects. An object can be better understood when viewed through multiple lenses." (NCTM, 2000, p. 360)
Watch the video segment (duration 0:27) in the viewer box on the upper left to hear a reflection from Pam Hardaway, a middle school teacher in California. Her ideas about manipulative materials are applicable in grades 3-5 as well.
Students benefit from a balanced instructional program in which they grapple with their own interpretation of the use of a particular representation, participate in discussions, and share ideas with their teacher and peers. This can lead to learning about common conventions of representations, and experimenting with uses of representations as suggested by others. In addition, as students compare and discuss connections between representations, they deepen their understanding of mathematical ideas.
As a follow-up to the Representing Data activity from Part A, the teacher asked a student, Cody, to gather data on the weight of pennies. She was hoping to bring up connections to previous science work with scales and to the use of decimals to report measures. Cody used the classroom scale and created a table as he weighed different numbers of pennies. He chose to use a coordinate graph to show his data. Because Cody had a large range in his data, the teacher guided him in his choice of scale and the labels for his graph. She also helped him understand that some variation in his numbers was to be expected when using real pennies and a real scale.
Here is Cody's data:
pennies graph
Later, while discussing Cody's table and the graph with the whole class, the teacher was able to extend the lesson to include algebraic representation. Several students were able to look at the line on the coordinate graph and, with the help of the teacher's questions, notice that the number of grams steadily increased. The class stated this observation in a verbal rule: "Every time you have 10 more pennies, you get about 25 more grams."
The teacher continued the discussion and asked about the general relationship between the data in the Number of Pennies row and the data for the corresponding number of grams. Several students said, "It's times two and a half!", and the class went on to record another rule: "The number of grams is (about) 2.5 times as much as the number of pennies." They then shortened their rule to "g = 2.5p" by using the letter g to stand for the number of grams and the letter p to stand for the number of pennies. This equation is an example of an algebraic symbolic expression, a linear equation showing constant growth.
The discussion also included acknowledgement that this formula doesn't perfectly predict the weight of a number of pennies. For example, 100 pennies only weighed 248 grams, not 250. Their teacher showed them the "is about equal to" symbol and wrote "g ≈ 2.5p" before returning to the central purpose of the lesson: developing connections between various forms of representation for relationships in sets of data.
Notice that the development of the equation was built on a series of translations among various representations. The situation was used to generate a table of values, related points were plotted on a coordinate grid, and a verbal rule was stated. The rule was then translated into a symbolic form.
Through such lessons, upper elementary students gain understanding that will support their success with more formal algebraic ideas and skillful use of representations during the middle grades.
Teacher's Role in fostering representations
Students are frequently advised to "Draw a picture," "Use models," or "Write a number sentence." Students further develop their own problem-solving skills and their understanding of operations and procedures by using a variety of representations. However, the power of representation is greatly enhanced when the right balance is achieved between independent exploration, reflection on the work of others, and experiences with new types of representation that build upon each other.
The teacher can model the use of a variety of representations as a class or in a group in order to solve problems. By observing students as they work on rich problems, the teacher also has an opportunity to select a number of students or groups to share their representation and thinking. Through careful questioning, the similarities and differences, as well as the general usefulness, of different representations can be discovered.
Certain conventional tools of representation, such as number lines, bar graphs, and coordinate graphs, have important design elements that must be explicitly brought to the attention of all students. Other tools are more open organizational devices and can be developed over time through problem solving and comparisons of methods. However, "many students need support in constructing pictures, graphs, tables, and other representations. If they have many opportunities for using, developing, comparing, and analyzing a variety of representations, students will become competent in selecting what they need for a particular problem" (NCTM, 2000, p. 208).
It is beneficial to work from different starting points, which the teacher can help foster. For example, students might be given the equation n = 6(x); their task is to make up a situation that fits this equation and then to model it with manipulative materials. Similarly, a table of values might be given, and the situation and labels for the first and second columns will then be decided on by a class or small group of students.
Here is what one student group came up with for the n = 6(x) equation:
Table of Values?
Selecting tasks is one of a teacher's most critical responsibilities of the teacher. Rich mathematical tasks that are compatible with a variety of representations offer students opportunities to make decisions as to which form of representation to use, to see other representations used and discussed, to make connections between representations, and to practice the use of new forms.
Representations are truly powerful thinking tools for upper elementary students. Through work with a variety of manipulative materials, diagrams, graphs, and equations, students can develop lasting understanding of and skill with important mathematical topics while also building strong foundations for future learning. By looking for connections among representations, students are able to continue to see mathematics as a unified, sensible subject area. Representations also facilitate communication with others and serve as records of information. In the course of solving and discussing rich problems under a teacher's guidance, all students can expand their use of representations.
"Mathematical representation" refers to the wide variety of ways to capture an abstract mathematical concept or relationship. A mathematical representation may be visible, such as a number sentence, a display of manipulative materials, or a graph, but it may also be an internal way of seeing and thinking about a mathematical idea. Regardless of their form, representations can enhance students' communication, reasoning, and problem-solving abilities; help them make connections among ideas; and aid them in learning new concepts and procedures.
"Good representations fulfill a dual role: they are tools for thinking and instruments for communicating . . . Learning to record or represent thinking in an organized way, both in solving a problem and in sharing a solution, is an acquired skill for many students. Teachers can and should emphasize the importance of representing mathematical ideas in a variety of ways."
(NCTM, 2000, p. 206, 208)
Defining Representation
"The ways in which mathematical ideas are represented is fundamental to how people can understand and use those ideas . . . When students gain access to mathematical representations and the ideas they represent, they have a set of tools that significantly expand their capacity to think mathematically."
(NCTM, 2000, p. 67)
Representations include symbols, equations, words, pictures, tables, graphs, manipulative objects, and actions as well as mental, internal ways of thinking about a mathematical idea. Representations are powerful thinking tools. However, for many students, this power is not accessible unless they receive purposeful guidance in expanding their repertoire.
The act of representing a concept or relationship may result in the use of manipulative materials, the construction of graphs or diagrams, the writing of number sentences, or the presentation of a written or oral explanation. When using representations to solve a problem or make sense of a new concept, students are likely to go back and forth, using the representation to help clarify the problem and using the problem to extend their understanding of the representation.
"Representations do not 'show' the mathematics to the students. Rather, the students need to work with each representation extensively in many contexts as well as move between representations in order to understand how they can use a representation to model mathematical ideas and relationships." (NCTM, 2000, p. 208)
Using Variety of Representations
Much of students' mathematical learning involves expanding understanding of a mathematical idea or relationship by shifting from one type of representation to a different representation of the same relationship. This is one of the reasons that it is important for students to use a variety of manipulative materials, which are then carefully related to paper-and-pencil methods of solving problems. Through this work, they move from informal representations to the more formal, and abstract representations that more advanced work will require.
A third-grade teacher introduced a unit of study on two-digit multiplication with an open-ended assignment for student pairs. The students were asked to think of a story problem where someone would want to know the product of 15 x 12 and to then show a method for finding that product:
Students suggested several different contexts and manipulative materials to be used. For example, one suggestion was 15 plastic bags with 12 crayons each (represented by 15 rectangles with "12 crayons" written on each rectangle). Another suggestion was a dot array of 15 teams lined up on the playground, with 12 players on each team. Another group of students worked first on their solution method and struggled to match 15 x 12 to a story problem. They used a method that had been used to introduce multiplication in the prior grade: making 15 towers of linking cubes with 12 cubes each and then finding an efficient method for counting them all:
The methods used for finding the product were somewhat dependent on the representation that was used. The group with the crayons added 12 plus 12 plus 12, etc. -- first mentally, then on paper -- to find the total; they had some difficulty keeping track of the number of times that 12 was added:
The two groups with the array of team members and with the towers of linking cubes both experimented with a variety of ways of finding the product before partitioning their rows of dots or towers of cubes into two parts, 10 and 2. They worked first with the rows or towers of 10, finding that partial product easily, and then found the partial product of 15 times 2.
As students invent their own way to show a relationship, such as 15 x 12, with materials, pictures, or diagrams, they engage in thought that helps strengthen their understanding of the operation of multiplication. When a number of different representations for a given problem are shared and discussed, similarities in the mathematical structure of each representation can be highlighted. For example, a student with 15 towers with 12 linking cubes in each tower can point out the similarity to a representation that uses 15 plastic bags with 12 crayons in each. Similarly, an area diagram can be related to an array that is made of 12 rows with 15 objects per row:
Notice that while an array representation may initially encourage students to use simple counting to solve the problem, an area model clearly shows the advantage of making smaller, easier to calculate groups or areas. This also very clearly corresponds to the standard algorithm for multiplication, as shown above.
During the following week, the teacher extended the example of the lines of team players and connected it to using base-ten blocks to represent 1, 10, or 100 players instead of drawing individual dots. Over the course of the next several weeks, through class discussion and guidance from the teacher, the class developed connections between this manipulative model, arrays drawn on grid paper, and symbolic methods for finding the product of two two-digit numbers. They also practiced mental-math methods for finding products by breaking a problem into two parts, such as (15 x 10) + (15 x 2).
"[D]ifferent representations support different ways of thinking about and manipulating mathematical objects. An object can be better understood when viewed through multiple lenses." (NCTM, 2000, p. 360)
Watch the video segment (duration 0:27) in the viewer box on the upper left to hear a reflection from Pam Hardaway, a middle school teacher in California. Her ideas about manipulative materials are applicable in grades 3-5 as well.
Students benefit from a balanced instructional program in which they grapple with their own interpretation of the use of a particular representation, participate in discussions, and share ideas with their teacher and peers. This can lead to learning about common conventions of representations, and experimenting with uses of representations as suggested by others. In addition, as students compare and discuss connections between representations, they deepen their understanding of mathematical ideas.
As a follow-up to the Representing Data activity from Part A, the teacher asked a student, Cody, to gather data on the weight of pennies. She was hoping to bring up connections to previous science work with scales and to the use of decimals to report measures. Cody used the classroom scale and created a table as he weighed different numbers of pennies. He chose to use a coordinate graph to show his data. Because Cody had a large range in his data, the teacher guided him in his choice of scale and the labels for his graph. She also helped him understand that some variation in his numbers was to be expected when using real pennies and a real scale.
Here is Cody's data:
Later, while discussing Cody's table and the graph with the whole class, the teacher was able to extend the lesson to include algebraic representation. Several students were able to look at the line on the coordinate graph and, with the help of the teacher's questions, notice that the number of grams steadily increased. The class stated this observation in a verbal rule: "Every time you have 10 more pennies, you get about 25 more grams."
The teacher continued the discussion and asked about the general relationship between the data in the Number of Pennies row and the data for the corresponding number of grams. Several students said, "It's times two and a half!", and the class went on to record another rule: "The number of grams is (about) 2.5 times as much as the number of pennies." They then shortened their rule to "g = 2.5p" by using the letter g to stand for the number of grams and the letter p to stand for the number of pennies. This equation is an example of an algebraic symbolic expression, a linear equation showing constant growth.
The discussion also included acknowledgement that this formula doesn't perfectly predict the weight of a number of pennies. For example, 100 pennies only weighed 248 grams, not 250. Their teacher showed them the "is about equal to" symbol and wrote "g ≈ 2.5p" before returning to the central purpose of the lesson: developing connections between various forms of representation for relationships in sets of data.
Notice that the development of the equation was built on a series of translations among various representations. The situation was used to generate a table of values, related points were plotted on a coordinate grid, and a verbal rule was stated. The rule was then translated into a symbolic form.
Through such lessons, upper elementary students gain understanding that will support their success with more formal algebraic ideas and skillful use of representations during the middle grades.
Teacher's Role in fostering representations
Students are frequently advised to "Draw a picture," "Use models," or "Write a number sentence." Students further develop their own problem-solving skills and their understanding of operations and procedures by using a variety of representations. However, the power of representation is greatly enhanced when the right balance is achieved between independent exploration, reflection on the work of others, and experiences with new types of representation that build upon each other.
The teacher can model the use of a variety of representations as a class or in a group in order to solve problems. By observing students as they work on rich problems, the teacher also has an opportunity to select a number of students or groups to share their representation and thinking. Through careful questioning, the similarities and differences, as well as the general usefulness, of different representations can be discovered.
Certain conventional tools of representation, such as number lines, bar graphs, and coordinate graphs, have important design elements that must be explicitly brought to the attention of all students. Other tools are more open organizational devices and can be developed over time through problem solving and comparisons of methods. However, "many students need support in constructing pictures, graphs, tables, and other representations. If they have many opportunities for using, developing, comparing, and analyzing a variety of representations, students will become competent in selecting what they need for a particular problem" (NCTM, 2000, p. 208).
It is beneficial to work from different starting points, which the teacher can help foster. For example, students might be given the equation n = 6(x); their task is to make up a situation that fits this equation and then to model it with manipulative materials. Similarly, a table of values might be given, and the situation and labels for the first and second columns will then be decided on by a class or small group of students.
Here is what one student group came up with for the n = 6(x) equation:
Selecting tasks is one of a teacher's most critical responsibilities of the teacher. Rich mathematical tasks that are compatible with a variety of representations offer students opportunities to make decisions as to which form of representation to use, to see other representations used and discussed, to make connections between representations, and to practice the use of new forms.
Representations are truly powerful thinking tools for upper elementary students. Through work with a variety of manipulative materials, diagrams, graphs, and equations, students can develop lasting understanding of and skill with important mathematical topics while also building strong foundations for future learning. By looking for connections among representations, students are able to continue to see mathematics as a unified, sensible subject area. Representations also facilitate communication with others and serve as records of information. In the course of solving and discussing rich problems under a teacher's guidance, all students can expand their use of representations.